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In this paper, we prove the complete monotonicity of some functions involving Bateman's G--function and show that ... where α = 1 and β = 0 are the best possible constants, which is a refinement of a recent result. Then, we give a new proof of Slavic inequality about Wallis ratio Wm and provide a new inequality for Wm. Our new inequality improves some recent related works. We also present two inequalities for the hyperbolic tangent function. [ABSTRACT FROM AUTHOR]

In the paper, we presented a family M(μ x) of approximations of the Bateman function G(x). The family M(μ x) = G(x) for a certain μ whenever x is fixed and it presented asymptotical approximation of the Bateman's G-function as x→1. We studied the order of convergence of the approximations M(μ x) of the function G(x). Some properties and bounds of the error are deduced. We presented new sharp double inequality of G(x) with the upper and lower bounds M(1; x) and M( 4 /e2-4 ; x) (resp.). Also, we show that the approximations M(μ; x) are better than the approximation 1/x + 1/2x2 for any μ in an open subinterval of [ 1, 4/e2-4 ]. [ABSTRACT FROM AUTHOR]

In the paper, we presented a family M(μ, x) of approximations of the Bateman function G(x). The family M(μ, x) = G(x) for a certain μ whenever x is fixed and it presented asymptotical approximation of the Bateman's G-function as x → ∞. We studied the or- der of convergence of the approximations M(μ, x) of the function G(x). Some properties and bounds of the error are deduced. We presented new sharp double inequality of G(x) with the upper and lower bounds M(1; x) and M( 4/e2-4, x) (resp.). Also, we show that the approximations M(μ, x) are better than the approximation 1/x + 12x2 for any μ in an open subinterval of [1; 4/e2-4]. [ABSTRACT FROM AUTHOR]

In the paper, the authors established two best approximation formulas for the Bateman's G-function. Also, they studied the completely monotonicity of some functions involving G(x). Some new inequalities are deduced for the function and its derivative such as 1/2 ln [1 + 2x + a/x2 + 2x + 4/3 ] < G(x + 2) < 1/2 ln [ 1 + 2x + b/x2 + 2x + 4/3 ], x > 0 where a = 3 and b = e4-16/12 are the best possible constants. Our results improve some recent inequalities about the function G(x). [ABSTRACT FROM AUTHOR]

In the paper, the authors established two best approximation formulas for the Bateman's G-function. Also, they studied the completely monotonicity of some functions involving G(x). Some new inequalities are deduced for the function and its derivative such as 1/2 ln [1 + 2x + a/x2 + 2x + 4/3 ] < G(x + 2) < 1/2 ln [ 1 + 2x + b/x2 + 2x + 4/3 ], x > 0 where a = 3 and b = e4-16/12 are the best possible constants. Our results improve some recent inequalities about the function G(x). [ABSTRACT FROM AUTHOR]